Simultaneous stabilization in $A_{\mathbb R}(\mathbb D)$
Tom 191 / 2009
Studia Mathematica 191 (2009), 223-235
MSC: Primary 46E25; Secondary 46J15, 93D99.
DOI: 10.4064/sm191-3-4
Streszczenie
We study the problem of simultaneous stabilization for the algebra $A_\mathbb R(\mathbb D)$. Invertible pairs $(f_j,g_j)$, $j=1,\ldots, n$, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair $(\alpha,\beta)$ of elements such that $\alpha f_j+\beta g_j$ is invertible in this algebra for $j=1,\ldots, n$.
For $n=2$, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_\mathbb R(\mathbb D)$ has stable rank two, we are faced here with a different situation. When $n=2$, necessary and sufficient conditions are given so that we have simultaneous stability in $A_\mathbb R(\mathbb D)$.
For $n\geq 3$ we show that under these conditions simultaneous stabilization is not possible and further connect this result to the question of which pairs $(f,g)$ in $A_\mathbb R(\mathbb D)^2$ are totally reducible, that is, for which pairs there exist two units $u$ and $v$ in $A_\mathbb R(\mathbb D)$ such that $uf+vg=1$.
